Geodesic
Structure of $k$-closures of finite nilpotent permutation groups
Algebra i logika, Tome 60 (2021) no. 2, pp. 231-239

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Let $G$ be a permutation group of a set $\Omega$ and $k$ be a positive integer. The $k$-closure of $G$ is the greatest (w.r.t. inclusion) subgroup $G^{(k)}$ in $\mathrm{Sym} (\Omega)$ which has the same orbits as has $G$ under the componentwise action on the set $\Omega^k$. It is proved that the $k$-closure of a finite nilpotent group coincides with the direct product of $k$-closures of all of its Sylow subgroups.
Keywords: $k$-closure, finite nilpotent group, Sylow subgroup. 
@article{AL_2021_60_2_a7,
     author = {D. V. Churikov},
     title = {Structure of $k$-closures of finite nilpotent permutation groups},
     journal = {Algebra i logika},
     pages = {231--239},
     publisher = {mathdoc},
     volume = {60},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2021_60_2_a7/}
}
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D. V. Churikov. Structure of $k$-closures of finite nilpotent permutation groups. Algebra i logika, Tome 60 (2021) no. 2, pp. 231-239. http://geodesic.mathdoc.fr/item/AL_2021_60_2_a7/